INVARIANT FORM OF COULOMB CORRECTIONS IN THE THEORY OF NONLINEAR IONIZATION OF ATOMS BY INTENSE LASER RADIATION
S. V. Popruzhenko*
National Research Nuclear University MEPhI II54O9, Moscow, Russia
Received November 24, 2013
Using the imaginary time method, a new formulation of Coulomb corrections to the amplitude of nonlinear ionization of atoms is given. The Coulomb corrections to the photoelectron action and trajectory are presented in the form independent of the integration path in the imaginary time plane. The obtained representation corrects the previously known results and shows that the subdivision of photoelectron motion into the sub-barrier and after-barrier parts is conditional and does not influence observables. The new correction is particularly relevant in the multiphoton regime of ionization.
DOI: 10.7868/S0044451014040089
1. INTRODUCTION
The theory of nonlinear ionization of atoms by intonso laser radiation originates in the seminal work of Keldysh fl], whore an efficient nonperturbative approximation for the amplitude of ionization by an intense low-frequency electromagnetic field was formulated. The term "low-frequency field" means in this content that the ionization potential Ip of an atom is much greater than the photon energy ñu), i. e., the 11ml-tiquantum parameter is large,
A'o = Ip/u > 1.
(1)
Under this condition, ionization can only proceed via a nonlinear mechanism. The Keldysh ionization ansatz can be summarized as follows. In a strong laser field, the electron continuum states can be with reasonable accuracy approximated by Gordon Volkov waves [2, 3], solutions of the Schrodinger equation (Klein Gordon or Dirac equation in the relativistic case) for an electron in the field of a plane electromagnetic wave. If the laser field is strong enough, the interaction of a liberated electron with its parent ion can be disregarded in the zeroth approximation. On the other hand, in order to fully ionize a bound atomic level, the electric field strength io well below the characteristic electric field
£ck of this level is usually sufficient, and therefore the condition
F = £0/£ch « 1
(2)
is satisfied for most of the cases. Hero, the characteristic field is defined as
3/2
£ch =
m" eh
2Ij,
m
(3)
with m and e being the electron IIlclSS and the elementary charge. Under conditions (1) and (2), the influence of the laser field 011 the bound state can be disregarded, and the ionization amplitude can be presented in the form
MP) = -j: J d'ix^v)(x)Vini(x)^h(x), (4)
where is the bound field-free atomic state, <I'p is the Volkov function corresponding to the asymptotic electron momentum equal to p, and I",,,/ is the electron field interaction operator. Equation (4) gives the probability amplitude of nonlinear ionization at arbitrary values of the Keldysh parameter
7 =
\ßmïp
UJ
e£n
(5)
E-mail: sergey.popruzhenko'ögmail.com
where ui is the laser-field frequency.
The theoretical approach based 011 the above idea is known in the literature as the Keldysh theory or
strong field approximation (SFA) [4, 5]. Over the time after publication of Keldysh's work fl], it was essentially developed and applied for description of a variety of strong-field phenomena. The present status of the Keldysh theory and SFA was reviewed in Refs. [6 8].
Amplitude (4) does not account for the electron ion interaction in the continuum. For systems bound by-short-range forces, e.g., negative ions, this interaction is small, but still causes observable effects. For a review of theoretical approaches to description of strong-field ionization of negative ions, we refer the reader to [9, 10] and the references therein. For atoms and molecules, the Coulomb force generates significant effects, whose description requires an essential modification of the theory. This was achieved by the introduction of Coulomb corrections into the phase of amplitude (4). Evaluation of these corrections is based on the imaginary time method (ITM) [11], which allows expressing amplitude (4) via the electron action in the field of a plane electromagnetic wave, calculated along a classical trajectory in complex time. In the early paper by Perelomov and Popov [12], the ITM was applied for calculation of the total ionization rate of atoms in the tunneling limit, 7 C 1. It was shown there that the Coulomb field enhances the rate of ionization typically by several orders of magnitude. Later, this result was generalized to the case of arbitrary values of the Keldysh parameter [13] (assuming that inequality (1) is satisfied, however). Besides enhancing the total ionization rate, the Coulomb interaction was shown to generate several effects accessible for experimental observation, including the Coulomb asymmetry in elliptically polarized fields [14 16], cusps and double-hump structures [17 19], low-energy structures [20 23], and side lobes [24] in momentum spectra of photoelectrons.
Currently, the method of Coulomb corrections in the theory of strong-field ionization is well developed. This includes classical trajectory simulations, a rela-tivistic version of the Keldysh theory, the trajectory-based SFA, and other approaches. For details, we refer to [6,8,25] and the references therein. The aim of this paper is to address one controversial issue inherent to all the above-mentioned methods of evaluation of Coulomb corrections. Namely, the calculation procedure involves the photoelectron tunnel exit a spatial point where the electron appears in real time and space after ionization. In purely classical simulations (e.g., in Refs.[15, 21, 23]), the tunnel exit is a starting point for a calculation, and the influence of the electron ion interaction 011 the electron dynamics before the electron appears at the exit is not considered. In quantum mechanical calculations, including
the ITM, both sub-barrier and after-barrier motion of the electron are taken into account. The sub-barrier motion mostly yields the imaginary part of the action and influences the absolute value of the ionization probability. Photoelectron motion after the barrier proceeds in real time and space and influences the real part of the action, and therefore the interference structure of photoelectron spectra. As a result, the tunnel exit enters Coulomb corrections to the photoelectron action. On the other hand, the position of the tunnel exit is not an observable, and hence it must not influence momentum distributions.
The question therefore arises: is it possible to formulate the method of Coulomb corrections in a form that does not involve the tunnel exit, but only depends 011 the observables of the problem? Such a formulation is given in this paper. It is shown that the Coulomb correction to the photoelectron momentum can be presented in the form of a converging integral in the complex time plane, which only depends 011 the momentum itself. The integration paths must be chosen taking the analyticity properties of the Coulomb interaction energy in complex space into account. It is then shown that the obtained Coulomb correction reproduces the previously known result in the tunneling regime 7 -C 1, but this is not the case for 7 1.
This paper is organized as follows. In Sec. 2, we introduce basic equations and briefly describe the standard approach to the calculation of Coulomb corrections. In Sec. 3, we derive an invariant form of the Coulomb correction to the photoelectron momentum, which does not involve the tunnel exit position. The choice of the integration path in the complex time plane is then discussed. The last section contains conclusions.
2. BASIC EQUATIONS
Using the ITM, ionization amplitude (4) can be represented in the form (here and hereafter, atomic units fi, = rn = e are used) [11,12]
.40(p) xexp(¿Mo(p,ís,T)),
(6)
where W"0 is the reduced electron action in the laser field £(/).
W"o(p, ts(p),T) =
1
/ {~ ' r° ~ Ip]dt ~ r° ' V°
(7)
and the trajectory r0(t) satisfies the Newton equation
ro = -£(t), (8)
with the initial and final conditions
v2(f„)=-2Ipi ro(ig) = 0, v0(T) = p. (9)
Here, T —¥ oc is the time instant when the electron, having the velocity v0 and momentum p is observed at the detector. A preexponential factor not important for our purposes is omitted in (6). Its particular form is determined by the initial-state wave function [6, 7, 26]. The first equation in (9) shows that the initial time t8 of electron motion is always complex, while the second equation assumes that before the ionization event, the electron was confined in the atom. Introducing the laser field vector potential such that £(t) = —A(f), we can represent the first equation in (9) in the form
[p + A(fg)]2 = —2IP
(10)
which determines a complex saddle point t8(p). The ITM equations provide us with a physically appealing picture of ionization: the electron starts from the origin at a complex time instant t8 = to + iro, having an imaginary initial velocity v0(t8) = ±isj2Ip. As time arrives to the real axis, t = to, the velocity also becomes real. The electron coordinate b = r0(io) is also real for the most probable trajectory that minimizes the imaginary part of action (7). This point b is interpreted as the tunnel exit. For a linearly polarized monochromatic field
£(t) = Eoc.osif, (f = u)t,
we obtain
'0
voM = P--- smip,
ul
r0(ip) = -(ip- ip„) + -£(cos<^> - COS ul ulz
(11)
(12)
For the most probable trajectory p = 0, ips = ¿Arcsli7, and the tunnel exit point is given by
b=£o_
u/2
1
• T
1
(13)
In the tunneling limit 7 -C 1, this gives the standard potential barrier width in a static field b = Ip/£0; in the opposite multiphoton regime, b = s/2Ip/^.
The procedure introducing Coulomb corrections to amplitude (6) is as follows [13, 25].
1. The Coulomb-free trajectory is replaced by a corrected one:
vo(i)=p+A(i)-»-vo(i)+vi(i)1 vi =
Zvo
(14)
where Z is the atomic residual charge (Z = 1 for ionization of neutral atoms and Z = 0 for negative ions).
2. The
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